More lessons from complexity E:CO Issue Vol. 8 No pp Forum

Transcription

1 More lessons from complexity E:CO Issue Vol. 8 No pp Forum More lessons from complexity - The origin: The root of peace Carlos E. Puente Department of Land, Air and Water Resources, University of California, USA The last few decades have witnessed the development of a host of ideas aimed at understanding and predicting nature s ever present complexity (see for instance, Mandlebrot, 1982; Bak, 1996; Wolfram, 2002). It is shown that such a work provides, through its detailed study of order and disorder, a suitable framework for visualizing the dynamics and consequences of mankind s ever present divisive traits. Specifically, this work explains how recent universal results pertaining to the transition from order to chaos via a cascade of bifurcations point us to a serene state, symbolized by the convergence to the origin in the root of a Feigenbaum s tree, in which we all may achieve peace. When α is increased up to a value α = , successive bifurcations in powers of two happen quickly, that is, the dynamics repeat exactly every 2 n generations, for any value of n; When α < α 4, behavior is found either periodic or non periodic. For instance, for α = 3.6 an infinite strange attractor with a whole in the middle is found (Figure 5); (1) (2) (3) When α = 3.83 = {X } and the dynamics oscillate every 3 generations (Figure 6); When α = 4, the most common behavior is non periodic and a dense strange attractor over the interval [0, 1] is found (Figure 7). The Logistic Map Recall the exotic dynamics of the logistic map (see for example, Rasband, 1990; Peitgen, et al., 1992; Beck & Schlögl, 1993): X k+1 = α X k (1 - X k ) that is, the chain of ultimately stable (and unstable) values X (α) found iterating the map, where X k denotes the normalized size of a population at generation k and α is a free parameter having values between 0 and 4: When α 1, the logistic parabola is below the one to one line (added to aid in the calculations), and then X = 0 (Figure 1); When 1 < α 3, the parabola is above the line X = Y and X = (α - 1)/α, the non-zero intersection between the curve and the straight line (Figure 2); (1) (2) When 3 < α = {X } and the population settles into an oscillation repeating every two generations (Figure 3); (3) When < α = {X, (4) (5) (6) X }. The population ultimately repeats every four generations, and the dynamics have experienced a bifurcation (Figure 4); Figure 1 Convergence to the origin Figure 2 Convergence to a fixed point Puente 115

2 Figure 3 Convergence to a 2-cycle. Figure 6 Convergence to a 3-cycle Figure 4 Convergence to a 4-cycle Figure 7 Convergence to a maximal non-repetitive set At the end, the cascade of stable perioddoubling bifurcations (before α ) and the emergence of chaos (strange attractors) intertwined with periodic behavior (including any period greater than two) is summarized via the celebrated Feigenbaum s diagram (Figure 8). This is so named after Mitchell Feigenbaum who showed that the bifurcation openings and their durations happen universally for a class of unimodal maps according to two universal constants F 1 and F 2, as follows (Feigenbaum, 1978) (refer to Figure 9): Figure 5 Convergence to a dusty non-repetitive attractor d n /d n+1 F 1 = , Δ n /Δ n+1 F 2 = E:CO Vol. 8 No pp

3 Figure 8 Bifurcations tree for the logistic map Figure 9 Successive bifurcations. X corresponds to supercycles, i.e., ½ for the logistic map. Puente 117

4 For example, other fig trees guided by F 1 and F 2 and for the two simple mappings f(x) = α X (1 X 3 ) and f(x) = α X (1 - X) 3 are shown below[1]. Notice how such contain: a straight root, a bent branch, bifurcation branches, and then, in an orderly intertwined fashion, following Sharkovskii s order (see for example, Rasband, 1990; Peitgen, et al., 1992; Beck & Schlögl, 1993), periodic branches, and the ever dusty foliage of chaos, where the unforgiving condition of sensitivity to initial conditions rules. Chaos theory and our quest for peace As the dynamics of the logistic map describe several physical processes (see for instance, Cvitanović, 1989; Bai-Lin. 1984), including fluid turbulence as induced by heating, that is, convection, it is pertinent to consider such a simple and universal mechanism to study how chaos and its related condition of violence may arise in the world. Given that the key parameter α, associated with the amount of heat (Libchaber & Maurer, 1978), dictates the ultimate organization of the fluid, we may see that it is wise to keep it small (in the world, and within each one of us) in order to avoid undesirable nonlinearities. For although the allegorical fig trees exhibit clear order in their pathway towards disorder, we may appreciate in the uneasy jumping on strange attractors (and also on periodic ones), the anxious and foolish frustration we often experience (so many times deterministically!) when we, by choosing to live in a hurry, travel from place to place to place in high heat without finding our root. In this spirit, the best solution for each one of us is to slow down altogether the pace of life, coming down the tree, so that by not crossing the main threshold X = Y, that is, by choosing α 1, we may surely live without turbulence and chaos in the Figure 10 Bifurcation diagrams for two simple nonlinear maps 118 E:CO Vol. 8 No pp

5 Figure 11 Orbits of the logistic map (α = 4) ending up at zero robust state symbolized by X = 0[2,3]. For there is a marked difference between a seemingly laminar condition as it happens through tangent bifurcations (see for example, Rasband, 1990; Peitgen, et al., 1992; Beck & Schlögl, 1993) and being truly at peace, for the former invariably contains dramatic bursts of chaos and ample intermittency (see for example, Rasband, 1990; Peitgen, et al., 1992; Beck & Schlögl, 1993). As zero, that is, converging to the origin, is identified as the desired state, it is sensible to realize that such an organization, a trivial solution for X (α), even if unstable, may be reached even when the worst chaos engulfs us (α = 4). For the precise dynamics of the pre-images of zero do not wander for ever in high heat, but rather find the way to the root through a delicate hopscotch by the middle[4] (Figure 11). For it is tragic indeed to oscillate for ever (Figure 12). And more tragic yet to be close to the point and miss it altogether forever[5] (Figure 13). For the butterfly effect, with all probability and contrary to the illusion that it provides us with options, leaves us irremediably trapped in dust. At the end, the emergence of the modern science of complexity helps us visualize our ancient Puente choices. It is indeed best for us to live in serenity and in a simple manner, not amplifying and hence heeding the voice. For only the conscious order of Love does not suffer the destiny of arrogant stubbornness that justly receives the same bad luck of a parabolic tree that did not have any fruit, the same one that with its tender branch(es) and budding leaves, also announces horrendous times, but also very good ones, times of joy and of friendship. Acknowledgments This piece is dedicated, in memoriam, to Fr. Rafael García Herreros, a humble man with a clear vision, whose special touch instilled in me the need to dream, in order to contribute to mend our chaotic world. 119

6 Figure 12 Some orbits of the logistic map (α = 4) leading to a 3-cycle Figure 13 Some orbits of the logistic map (α = 4) ending in a strange attractor 120 E:CO Vol. 8 No pp

7 FEIGENBAUM S PARABEL (Carlos E. Puente) In the confines of science majestically stands a tree, with all numerals in dance in emergent chaos to see. In the instance of a trance a good day I drew a link, and here it is, at a glance, the wisdom that I received. Foliage of disorder trapped in empty dust, jumps astir forever enduring subtle thrust. Crossing of the outset leaving faithful root, looming tender offset failing to yield fruit. Cascade of bifurcations, increasing heat within, inescapable succession of branches bent by wind. Sprouting of dynamics attracted to the strange, oh infinity reminding at the origin: the flame. In the midst of chaos there is a small gate leading to fine rest. In the midst of chaos there are loyal paths inviting to a dance. On top of the fig tree there is a key point that runs to the core. In the midst of chaos there is leaping game discerning the way. In the midst of chaos there is a fine well watering the brain. On top of the fig tree there is a clean frame that cancels the blame. On top of the fig tree there is mighty help that shelters from hell. In the midst of chaos, look it is there, in the midst of chaos, logistics in truth, in the midst of chaos, a clear faithful route, in the midst of chaos, leading to the root. On top of the fig tree, this is no delusion, on top of the fig tree, a sought needle s eye, on top of the fig tree, the symbol of wheat, on top of the fig tree, surrounded by weeds. Could it be, oh my friends, that science provides a rhyme?, for a rotten tree foretells the very advent of time. Could it be, oh how plain, that nature extends a call?, for old parable proclaims the crux in growing small. On top of the fig tree there is a clear light that averts a fright. Puente 121

8 References Bai-Lin, H. (1984). Chaos, World Scientific, ISBN Bak, P. (1996). How Nature Works, Copernicus, ISBN X (1999). Beck, C. and Schlögl, F. (1993). Thermodynamics of Chaotic Systems: An Introduction, Cambridge, UK: Cambridge University Press, ISBN (1995). Cvitanović, P. (ed.) (1989). Universality in Chaos, Adam Hilger, ISBN Feigenbaum, M. J. (1978). Quantitative universality for a class of nonlinear transformations, Journal of Statistical Physics, ISSN , 19(1): 25. Libchaber, A. and Maurer, J. (1978). Local probe in a Rayleigh-Benard experiment in liquid helium, J. de Phys. Lett. 39, L369. Mandelbrot, B. B. (1982). The Fractal Geometry of Nature, Freeman, ISBN Peitgen, H.-O., Jürgens, H., Saupe, D. (1992). Chaos and Fractals: New Frontiers of Science, New York, NY: Springer-Verlag, ISBN (2004). Puente, C. E. (2006a). Lessons from complexity: The hypotenuse - The pathway of peace, Emergence: Complexity & Organization, ISSN , 8(2): Puente, C. E. (2006b). The Hypotenuse: An Illustrated Scientific Parable for Turbulent Times, AuthorHouse, ISBN Rasband, S. N. (1990). Chaotic Dynamics of Nonlinear Systems, Chichester, UK: John Wiley & Sons, ISBN (1997). Wolfram, S. (2002). A New Kind of Science, Wolfram Media Inc., ISBN Notes [1] Feigenbaum means fig tree in German. [2] Of course, losing all our rabbits is quite undesirable, but surrendering to live slowly is a rather good investment. [3] Although it may be argued that the implied laminar flow is boring, this is the common ground that ensures proper communication amongst us. For α > 1, in its relentless amplification from generation to generation, corresponds to a proud state that prevents us from seeing with the eyes of somebody else. [4] The symbol of the (square) root may also be seen after two twists (left and up) of the graph! For additional insight on this concept consider Puente (2006a,b). [5] If tragedy may be quantified Choices Resting Surrendering Simplicity Order Serenity Peace To decrease Below X = Y Wandering Drifting Complexity Disorder Turbulence Chaos To increase Above X = Y 122 E:CO Vol. 8 No pp